Finite Group Databasebeta

← #30571064

Subgroups — #30571064

Order 15616.

1814 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459681C211yesmaximal, characteristic, complemented
2C2×#16840498C211yesmaximal, characteristic, complemented
3#21458833C214yesmaximal, complemented
4#21458819C214yesmaximal, complemented
5#21458520C214yesmaximal, complemented
6#21458483C214yesmaximal, complemented
7#21456781C214yesmaximal, complemented
8#21456760C211yesmaximal, characteristic, complemented
9#21455861C214yesmaximal, complemented
10#21455859C211yesmaximal, characteristic, complemented
11#21455846C211yesmaximal, characteristic, complemented
12#21454899C211yesmaximal, characteristic, complemented
13C61×#3205C211yesmaximal, characteristic
14#16840498—22no—
15#16840498—22no—
16#16840498C2214yes—
17#16840498C2214yes—
18#16840498C2212yescomplemented
19#16840487C2212yescomplemented
20#16840482C2212yescomplemented
21#16840481—22no—
22#16840469C2212yescomplemented
23#16840452—22no—
24#16840357—22no—
25#16840333—21no—
26#16840319C2211yescharacteristic
27#16840293C2214yes—
28C2×#14519764—21no—
29C2×#14519764C2212yes—
30C2×#14519758—22no—
31#16840245—21no—
32C2×C4×Dic122C2211yescharacteristic
33C2×C4×Dic122C2211yescharacteristic
34C2×D4×Dic61—21no—
35C2×D4×Dic61C2212yes—
36#16840145C2218yescomplemented
37#16840134C2212yes—
38#16840115C2218yescomplemented
39#16840108C2212yes—
40#16840098C2212yes—
41#16840078C2212yes—
42#16840060C2218yescomplemented
43#16840058C2212yescomplemented
44#16840044C2212yescomplemented
45#16839983C2218yescomplemented
46#16839975C2218yescomplemented
47#16839975C2218yescomplemented
48#16839973C2212yescomplemented
49#16839959C2214yes—
50#16839959C2212yescomplemented
51#16839920C2212yes—
52#16839913C2214yes—
53#16839900C2212yes—
54C8×Dic122C2214yescomplemented
55#16839886C2214yescomplemented
56C4×Dic244C2214yescomplemented
57#16839853C2218yescomplemented
58C4×#11758200C2214yescomplemented
59D8×Dic61C2218yescomplemented
60#16839815C2211yescharacteristic
61#16839812C2214yes—
62C61×#532C2211yescharacteristic
63C2×#127×C61C2211yescharacteristic
64C61×#493C2212yes—
65C61×#486C2212yes—
66C61×#459C2214yes—
67C61×#447C2214yes—
68C61×#431C2211yescharacteristic
69C22×Dic122D411yescharacteristic
70C22×Dic122C2311yescharacteristic
71C23×Dic61—22no—
72C2×Dic244C2314yes—
73C2×#11758200C2318yes—
74C2×#11758200C2314yes—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519775—22no—
78#14519770—24no—
79#14519770—24no—
80#14519769—21no—
81#14519768C2314yes—
82#14519767—24no—
83#14519764—22no—
84#14519764—24no—
85#14519764—24no—
86#14519764—24no—
87#14519764C2314yes—
88#14519764C2318yes—
89#14519758—24no—
90#14519758—24no—
91#14519758—28no—
92#14519758—28no—
93#14519756—21no—
94#14519756—22no—
95#14519756—22no—
96#14519756C2312yes—
97#14519754—24no—
98Q8×Dic61—22no—
99C4×Dic122—21no—
100C4×Dic122—21no—
101C4×Dic122—21no—
102C4×Dic122D412yes—
103C4×Dic122D412yes—
104C4×Dic122C2312yes—
105C4×Dic122C2312yes—
106C4×Dic122C2312yes—
107C4×Dic122C2312yes—
108C4×Dic122C2312yes—
109#14519742C2314yes—
110D4×Dic61—24no—
111D4×Dic61—24no—
112D4×Dic61—24no—
113D4×Dic61—22no—
114D4×Dic61—24no—
115D4×Dic61C2314yes—
116D4×Dic61C2318yes—
117#14519736—21no—
118#14519736—22no—
119#14519733—24no—
120C2×C4⋊Dic61—22no—
121C2×C4⋊Dic61D411yescharacteristic
122C2×C4⋊Dic61C2311yescharacteristic
123C2×#11758196—21no—
124C2×#11758196D412yes—
125C2×C22⋊Dic61—22no—
126C2×C22⋊Dic61—22no—
127C2×C22⋊Dic61—22no—
128C2×C4×Dic61—21no—
129C2×C4×Dic61C2312yes—
130#14519711C2314yes—
131#14519708C2314yes—
132C4⋊#92418C2312yes—
133#14519706C2318yes—
134#14519702C2314yes—
135#14519702C2314yes—
136#14519700C2314yes—
137#14519698C2318yes—
138#14519698C2314yes—
139C22⋊#92418C2312yes—
140C4×#92418C2312yes—
141C8×Dic61C2318yes—
142C2×#41×C61—21no—
143C22×D4×C61—21no—
144C22×D4×C61—21no—
145C22×D4×C61C2312yes—
146C2×D8×C61C2318yes—
147C4⋊Q8×C61C2311yescharacteristic
148#127×C61—21no—
149#127×C61—21no—
150#127×C61C2312yes—
151#127×C61C2312yes—
152#127×C61C2311yescharacteristic
153#124×C61—21no—
154#121×C61—21no—
155#121×C61C2312yes—
156C4×D4×C61—21no—
157C4×D4×C61C2312yes—
158C2×C42×C61C2311yescharacteristic
159C4⋊C8×C61C2312yes—
160#102×C61C2318yes—
161C22⋊C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C24×C61—22no—
164#41×C61—44no—
165C2×Q8×C61—21no—
166C2×D4×C61—22no—
167C2×D4×C61—21no—
168C2×D4×C61—22no—
169C2×D4×C61—22no—
170C2×D4×C61—22no—
171C2×D4×C61—21no—
172C2×D4×C61—22no—
173C2×D4×C61—22no—
174C2×D4×C61—44no—
175C2×D4×C61—44no—
176C2×D4×C61—82no—
177C2×D4×C61C2414yes—
178C2×D4×C61C2412yes—
179C22×C4×C61—22no—
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2411yescharacteristic
183C22×Dic61—21no—
184C22×Dic61—21no—
185C22×Dic61—28no—
186C22×Dic61—24no—
187C22×Dic61—84no—
188C22×Dic61C2×D412yes—
189C22×Dic61C2×D412yes—
190C2×Dic122—24no—
191C2×Dic122—21no—
192C2×Dic122—21no—
193C2×Dic122—42no—
194C2×Dic122C2×D412yes—
195C2×Dic122C2×D412yes—
196C2×Dic122C2412yes—
197C2×Dic122C2412yes—
198D8×C61—48no—
199C2×C8×C61C2414yes—
200C4⋊C4×C61—21no—
201C4⋊C4×C61C2412yes—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C42×C61C2×D411yescharacteristic
205C42×C61C2×D411yescharacteristic
206C42×C61C2411yescharacteristic
207C42×C61C2411yescharacteristic
208C22⋊Dic61—22no—
209C22⋊Dic61—28no—
210C22⋊Dic61—28no—
211C22⋊Dic61—28no—
212C22⋊Dic61—24no—
213C22⋊Dic61—24no—
214C22⋊Dic61—24no—
215#11758200—48no—
216#11758200—44no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—24no—
222C4⋊Dic61C2×D412yes—
223C4⋊Dic61C2×D412yes—
224C4⋊Dic61C2412yes—
225C4⋊Dic61C2412yes—
226#11758196—22no—
227#11758196—22no—
228#11758196—22no—
229#11758196—22no—
230#11758196—24no—
231#11758196—24no—
232#11758196C2×D414yes—
233#11758196C2×D414yes—
234C4×Dic61—22no—
235C4×Dic61—22no—
236C4×Dic61—22no—
237C4×Dic61—24no—
238C4×Dic61C2×D414yes—
239C4×Dic61C2414yes—
240C2×#92418C2414yes—
241Dic244—44no—
242C23×C61—22no—
243C23×C61—24no—
244C23×C61—44no—
245C23×C61—44no—
246C23×C61—44no—
247C23×C61C22×D411yescharacteristic
248Q8×C61—42no—
249D4×C61—24no—
250D4×C61—28no—
251D4×C61—44no—
252D4×C61—44no—
253D4×C61—44no—
254D4×C61—44no—
255D4×C61—42no—
256D4×C61—81no—
257D4×C61—82no—
258D4×C61—82no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—41no—
262C2×C4×C61—44no—
263C2×C4×C61#13514yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2511yescharacteristic
270C2×Dic61—24no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—84no—
279C2×Dic61—88no—
280C2×Dic61C22×D414yes—
281C2×Dic61C22×D414yes—
282Dic122—24no—
283Dic122—24no—
284Dic122—22no—
285Dic122—24no—
286Dic122—22no—
287Dic122—42no—
288Dic122—81no—
289C8×C61—24no—
290#92418—24no—
291#61885—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—44no—
295C22×C61—42no—
296C22×C61—44no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—44no—
300C22×C61—88no—
301C22×C61C2×#13512yes—
302C22×C61C23×D411yescharacteristic
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61C2×#13512yes—
308C4×C61C2×#13512yes—
309C4×C61#58512yes—
310Dic61—24no—
311Dic61—24no—
312Dic61—42no—
313Dic61—42no—
314Dic61—42no—
315Dic61—42no—
316#3507—611no—
317C2×#559—611no—
318#3396—614no—
319#3394—614no—
320#3365—614no—
321#3361—614no—
322#3211—614no—
323#3208—611no—
324#3205D6111yescharacteristic
325#3105—614no—
326#3104—611no—
327#3103—611no—
328#2982—611no—
329C2×C61—22no—
330C2×C61—82no—
331C2×C61—84no—
332C2×C61#348711yescharacteristic
333C2×C61C22×#13511yescharacteristic
334C2×C61#358511yescharacteristic
335#559—614no—
336#559—614no—
337#559—612no—
338#559—1222no—
339#559—1222no—
340#557—612no—
341#556—612no—
342#556—1222no—
343#554—612no—
344#552—1222no—
345#539—1222no—
346#536—1221no—
347#533—611no—
348#532C2×D6111yescharacteristic
349C2×#127C2×D6111yescharacteristic
350#529—614no—
351C2×#124—612no—
352C2×#124—1221no—
353C2×#123—1222no—
354#520—1221no—
355C2×C4×Q8—611no—
356C2×C4×Q8—611no—
357C2×C4×D4—612no—
358C2×C4×D4—1221no—
359#495—618no—
360#493C2×D6112yes—
361#492—612no—
362#488—618no—
363#486—612no—
364#486C2×D6112yes—
365#484—612no—
366#481—612no—
367#478—618no—
368#477—612no—
369#475—612no—
370#464—618no—
371#463—618no—
372#463—618no—
373#462—612no—
374#460—614no—
375#460—612no—
376#459C2×D6114yes—
377#451—612no—
378#450—614no—
379#448—612no—
380#447C2×D6114yes—
381C8×Q8—614no—
382C8×Q8—614no—
383C4×Q16—614no—
384C4×SD16—614no—
385C4×SD16—618no—
386C4×D8—618no—
387#431—611no—
388#431—614no—
389#431C2×D6111yescharacteristic
390C61#6188511yescharacteristic, complemented, Sylow
391C2×#41—21no—
392C22×Q8—611no—
393C22×Q8—611no—
394C22×D4—21no—
395C22×D4—21no—
396C22×D4C22×D6112yes—
397C23×C4—1222no—
398C2×Q16—614no—
399C2×SD16—614no—
400C2×SD16—618no—
401C2×D8C22×D6118yes—
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404C4⋊Q8—1222no—
405C4⋊Q8C22×D6111yescharacteristic
406#127—21no—
407#127—21no—
408#127C22×D6112yes—
409#127C22×D6112yes—
410#127C22×D6111yescharacteristic
411#126—1224no—
412#126—1224no—
413#125—614no—
414#125—1224no—
415#125—1221no—
416#124—21no—
417#124—618no—
418#124—614no—
419#124—1224no—
420#124—1224no—
421#124—1224no—
422#124—1222no—
423#123—1228no—
424#123—1228no—
425#123—1224no—
426#123—1224no—
427C22⋊Q8—612no—
428C22⋊Q8—1224no—
429C22⋊Q8—1222no—
430C22⋊Q8—1222no—
431C22⋊Q8—1221no—
432#121—21no—
433#121C22×D6112yes—
434C4×Q8—612no—
435C4×Q8—612no—
436C4×Q8—612no—
437C4×Q8—612no—
438C4×Q8—614no—
439C4×Q8—612no—
440C4×Q8—612no—
441C4×Q8—612no—
442C4×Q8—1222no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×Q8—1221no—
446C4×D4—21no—
447C4×D4—618no—
448C4×D4—614no—
449C4×D4—1224no—
450C4×D4—1224no—
451C4×D4—1224no—
452C4×D4—1222no—
453C4×D4—1224no—
454C4×D4C22×D6112yes—
455#117—1224no—
456#117—1222no—
457#117—1221no—
458C2×C4⋊C4—611no—
459C2×C4⋊C4—612no—
460C2×C4⋊C4—611no—
461C2×C4⋊C4—1222no—
462C2×C4⋊C4—1221no—
463C2×C22⋊C4—1222no—
464C2×C22⋊C4—1222no—
465C2×C22⋊C4—1222no—
466C2×C42—612no—
467C2×C42—1221no—
468C2×C42C22×D6111yescharacteristic
469#107—614no—
470#106—614no—
471C4⋊C8—618no—
472C4⋊C8—612no—
473C4⋊C8C22×D6112yes—
474#103—614no—
475#103—614no—
476#103—614no—
477#102—614no—
478#102—618no—
479#102C22×D6118yes—
480C22⋊C8—612no—
481C22⋊C8C22×D6112yes—
482C4×C8—618no—
483C4×C8—612no—
484C4×C8C22×D6112yes—
485C24—22no—
486#41—44no—
487C2×Q8—21no—
488C2×Q8—612no—
489C2×Q8—612no—
490C2×Q8—612no—
491C2×Q8—612no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1224no—
495C2×Q8—2442no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—21no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—44no—
505C2×D4—44no—
506C2×D4—82no—
507C2×D4C23×D6114yes—
508C2×D4C23×D6112yes—
509C22×C4—22no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—1224no—
513C22×C4—1228no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—4884no—
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520Q16—2444no—
521SD16—2444no—
522SD16—2448no—
523D8—48no—
524C2×C8—614no—
525C2×C8C23×D6114yes—
526C4⋊C4—21no—
527C4⋊C4—612no—
528C4⋊C4—614no—
529C4⋊C4—614no—
530C4⋊C4—612no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4C23×D6112yes—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—1228no—
548C22⋊C4—1228no—
549C22⋊C4—1228no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1224no—
553C22⋊C4—1222no—
554C42—614no—
555C42—614no—
556C42—1224no—
557C42—1222no—
558C42—1222no—
559C42—1222no—
560C42D4×D6111yescharacteristic
561C42D4×D6111yescharacteristic
562C42C23×D6111yescharacteristic
563C42C23×D6111yescharacteristic
564C23—22no—
565C23—24no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23C2×D4×D6111yescharacteristic
570Q8—42no—
571Q8—1224no—
572Q8—1224no—
573Q8—1224no—
574Q8—1222no—
575Q8—1222no—
576Q8—4881no—
577Q8—2442no—
578D4—24no—
579D4—28no—
580D4—44no—
581D4—44no—
582D4—44no—
583D4—44no—
584D4—42no—
585D4—81no—
586D4—82no—
587D4—82no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—614no—
593C2×C4—614no—
594C2×C4—1224no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—4884no—
600C2×C4—4888no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4#1451979614yes—
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C24×D6111yescharacteristic
611C8—24no—
612C8—1224no—
613C22—21no—
614C22—21no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—88no—
622C22C2×#1451979612yes—
623C22C22×D4×D6111yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—1224no—
629C4—1224no—
630C4—2442no—
631C4—2442no—
632C4—2442no—
633C4—2442no—
634C4C2×#1451979612yes—
635C4C2×#1451979612yes—
636C4#1684059712yes—
637C2—22no—
638C2—84no—
639C2—82no—
640C2#2145947511yescharacteristic
641C2C22×#1451979611yescharacteristic
642C2#2146059511yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
Fitting, prime radicalC61×#3205C2
socleC22×C61C23×D4
hypercenter#3205D61
hypocenterC61#61885
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21459681C2semidirect
2. C2×#16840498C2semidirect
3. #21458833C2semidirect
4. #21458819C2semidirect
5. #21458520C2semidirect
6. #21458483C2semidirect
7. #21456781C2semidirect
8. #21456760C2semidirect
9. #21455861C2semidirect
10. #21455859C2semidirect
11. #21455846C2semidirect
12. #21454899C2semidirect
18. #16840498C22semidirect
19. #16840487C22semidirect
20. #16840482C22semidirect
22. #16840469C22semidirect
36. #16840145C22semidirect
38. #16840115C22semidirect
42. #16840060C22semidirect
43. #16840058C22semidirect
44. #16840044C22semidirect
45. #16839983C22semidirect
46. #16839975C22semidirect
47. #16839975C22semidirect
48. #16839973C22semidirect
50. #16839959C22semidirect
54. C8×Dic122C22semidirect
55. #16839886C22semidirect
56. C4×Dic244C22semidirect
57. #16839853C22semidirect
58. C4×#11758200C22semidirect
59. D8×Dic61C22semidirect
390. C61#61885semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214596812
C2×#168404982
#214588332
#214588192
#214585202
#214584832
#214567812
#214567602
#214558612
#214558592
#214558462
#214548992
C61×#32052
#6188561